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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.expint.expint_i"></a><a class="link" href="expint_i.html" title="Exponential Integral Ei">Exponential Integral Ei</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.expint.expint_i.h0"></a>
        <span class="phrase"><a name="math_toolkit.expint.expint_i.synopsis"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.synopsis">Synopsis</a>
      </h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">expint</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">expint</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">expint</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
        The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
        type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
      </p>
<p>
        The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
        be used to control the behaviour of the function: how it handles errors,
        what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
        documentation for more details</a>.
      </p>
<h5>
<a name="math_toolkit.expint.expint_i.h1"></a>
        <span class="phrase"><a name="math_toolkit.expint.expint_i.description"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.description">Description</a>
      </h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">expint</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">expint</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
        Returns the <a href="http://mathworld.wolfram.com/ExponentialIntegral.html" target="_top">exponential
        integral</a> of z:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/expint_i_1.svg"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/expint_i.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.expint.expint_i.h2"></a>
        <span class="phrase"><a name="math_toolkit.expint.expint_i.accuracy"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.accuracy">Accuracy</a>
      </h5>
<p>
        The following table shows the peak errors (in units of epsilon) found on
        various platforms with various floating point types, along with comparisons
        to Cody's SPECFUN implementation and the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
        library. Unless otherwise specified any floating point type that is narrower
        than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
        zero error</a>.
      </p>
<div class="table">
<a name="math_toolkit.expint.expint_i.table_expint_Ei_"></a><p class="title"><b>Table 8.78. Error rates for expint (Ei)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for expint (Ei)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> long double
                </p>
              </th>
<th>
                <p>
                  GNU C++ version 7.1.0<br> linux<br> double
                </p>
              </th>
<th>
                <p>
                  Sun compiler version 0x5150<br> Sun Solaris<br> long double
                </p>
              </th>
<th>
                <p>
                  Microsoft Visual C++ version 14.1<br> Win32<br> double
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Exponential Integral Ei
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 5.05ε (Mean = 0.821ε)</span><br> <br>
                  (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 14.1ε (Mean = 2.43ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_expint_Ei___cmath__Exponential_Integral_Ei">And
                  other failures.</a>)
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.994ε (Mean = 0.142ε)</span><br> <br>
                  (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.96ε (Mean = 0.703ε))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 5.05ε (Mean = 0.835ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.43ε (Mean = 0.54ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Exponential Integral Ei: double exponent range
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.72ε (Mean = 0.593ε)</span><br> <br>
                  (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 3.11ε (Mean = 1.13ε))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 0.998ε (Mean = 0.156ε)</span><br> <br>
                  (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.5ε (Mean = 0.612ε))
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.72ε (Mean = 0.607ε)</span>
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.7ε (Mean = 0.66ε)</span>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Exponential Integral Ei: long exponent range
                </p>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.98ε (Mean = 0.595ε)</span><br> <br>
                  (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.93ε (Mean = 0.855ε))
                </p>
              </td>
<td>
              </td>
<td>
                <p>
                  <span class="blue">Max = 1.98ε (Mean = 0.575ε)</span>
                </p>
              </td>
<td>
              </td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
        It should be noted that all three libraries tested above offer sub-epsilon
        precision over most of their range.
      </p>
<p>
        GSL has the greatest difficulty near the positive root of En, while Cody's
        SPECFUN along with this implementation increase their error rates very slightly
        over the range [4,6].
      </p>
<p>
        The following error plot are based on an exhaustive search of the functions
        domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
        precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
        <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__double.svg" align="middle"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei__80_bit_long_double.svg" align="middle"></span>

        </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../graphs/exponential_integral_ei____float128.svg" align="middle"></span>

        </p></blockquote></div>
<h5>
<a name="math_toolkit.expint.expint_i.h3"></a>
        <span class="phrase"><a name="math_toolkit.expint.expint_i.testing"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.testing">Testing</a>
      </h5>
<p>
        The tests for these functions come in two parts: basic sanity checks use
        spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralEi" target="_top">Mathworld's
        online evaluator</a>, while accuracy checks use high-precision test values
        calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
        and this implementation. Note that the generic and type-specific versions
        of these functions use differing implementations internally, so this gives
        us reasonably independent test data. Using our test data to test other "known
        good" implementations also provides an additional sanity check.
      </p>
<h5>
<a name="math_toolkit.expint.expint_i.h4"></a>
        <span class="phrase"><a name="math_toolkit.expint.expint_i.implementation"></a></span><a class="link" href="expint_i.html#math_toolkit.expint.expint_i.implementation">Implementation</a>
      </h5>
<p>
        For x &lt; 0 this function just calls <a class="link" href="expint_n.html" title="Exponential Integral En">zeta</a>(1,
        -x): which in turn is implemented in terms of rational approximations when
        the type of x has 113 or fewer bits of precision.
      </p>
<p>
        For x &gt; 0 the generic version is implemented using the infinite series:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/expint_i_2.svg"></span>

        </p></blockquote></div>
<p>
        However, when the precision of the argument type is known at compile time
        and is 113 bits or less, then rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
        by JM</a> are used.
      </p>
<p>
        For 0 &lt; z &lt; 6 a root-preserving approximation of the form:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/expint_i_3.svg"></span>

        </p></blockquote></div>
<p>
        is used, where z<sub>0</sub> is the positive root of the function, and R(z/3 - 1) is
        a minimax rational approximation rescaled so that it is evaluated over [-1,1].
        Note that while the rational approximation over [0,6] converges rapidly to
        the minimax solution it is rather ill-conditioned in practice. Cody and Thacher
        <a href="#ftn.math_toolkit.expint.expint_i.f0" class="footnote" name="math_toolkit.expint.expint_i.f0"><sup class="footnote">[11]</sup></a> experienced the same issue and converted the polynomials into
        Chebeshev form to ensure stable computation. By experiment we found that
        the polynomials are just as stable in polynomial as Chebyshev form, <span class="emphasis"><em>provided</em></span>
        they are computed over the interval [-1,1].
      </p>
<p>
        Over the a series of intervals <span class="emphasis"><em>[a, b]</em></span> and <span class="emphasis"><em>[b,
        INF]</em></span> the rational approximation takes the form:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/expint_i_4.svg"></span>

        </p></blockquote></div>
<p>
        where <span class="emphasis"><em>c</em></span> is a constant, and <span class="emphasis"><em>R(t)</em></span>
        is a minimax solution optimised for low absolute error compared to <span class="emphasis"><em>c</em></span>.
        Variable <span class="emphasis"><em>t</em></span> is <code class="computeroutput"><span class="number">1</span><span class="special">/</span><span class="identifier">z</span></code> when
        the range in infinite and <code class="computeroutput"><span class="number">2</span><span class="identifier">z</span><span class="special">/(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span> <span class="special">-</span>
        <span class="special">(</span><span class="number">2</span><span class="identifier">a</span><span class="special">/(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span>
        <span class="special">+</span> <span class="number">1</span><span class="special">)</span></code> otherwise: this has the effect of scaling
        z to the interval [-1,1]. As before rational approximations over arbitrary
        intervals were found to be ill-conditioned: Cody and Thacher solved this
        issue by converting the polynomials to their J-Fraction equivalent. However,
        as long as the interval of evaluation was [-1,1] and the number of terms
        carefully chosen, it was found that the polynomials <span class="emphasis"><em>could</em></span>
        be evaluated to suitable precision: error rates are typically 2 to 3 epsilon
        which is comparable to the error rate that Cody and Thacher achieved using
        J-Fractions, but marginally more efficient given that fewer divisions are
        involved.
      </p>
<div class="footnotes">
<br><hr style="width:100; text-align:left;margin-left: 0">
<div id="ftn.math_toolkit.expint.expint_i.f0" class="footnote"><p><a href="#math_toolkit.expint.expint_i.f0" class="para"><sup class="para">[11] </sup></a>
          W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations for
          the exponential integral E<sub>1</sub>(x), Math. Comp. 22 (1968), 641-649, and W.
          J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential
          integral Ei(x), Math. Comp. 23 (1969), 289-303.
        </p></div>
</div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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